AP Calculus BC Formulas & Concepts

Limit Comparison Test

• If lim(n→∞) a_n/b_n is finite and positive, then ∑a_n and ∑b_n behave the same (both converge or diverge)

• If lim(n→∞) a_n/b_n is zero, then ∑a_n converges and ∑b_n diverges

• If lim(n→∞) a_n/b_n is infinite, then ∑a_n converges and ∑b_n diverges

• If lim(n→∞) a_n/b_n is finite and negative, then ∑a_n and ∑b_n behave the same (both converge or diverge)

Horizontal and Vertical Tangents

• Horizontal: f'(x) = 0 (numerator = 0)

• Horizontal: f'(x) DNE (denominator = 0)

• Vertical: f'(x) = 0 (numerator = 0)

• Vertical: f'(x) DNE (numerator = 0)

What is the definition of a Horizontal Asymptote?

• Let \( y = b \)

• Let \( y = ax + b \)

• Let \( y = \frac{1}{x} \)

• Let \( y = x^2 + b \)

Odd Function

• Symmetric over origin

  f(-x) = -f(x)

• Symmetric over the y-axis

  f(-x) = f(x)

• Has a maximum point

  f'(x) = 0

• Always increasing

  f'(x) > 0

Definition of a Limit

• @@\lim_{x\rightarrow a}f\left(x\right)@@ if and only if @@\lim_{x\rightarrow a^-}f\left(x\right)=\lim_{x\rightarrow a^+}f\left(x\right)=L@@

• @@\lim_{x\rightarrow a}f\left(x\right)@@ if and only if @@\lim_{x\rightarrow a^-}f\left(x\right)\neq\lim_{x\rightarrow a^+}f\left(x\right)@@

• @@\lim_{x\rightarrow a}f\left(x\right)@@ exists if and only if @@f(a) = L@@

• @@\lim_{x\rightarrow a}f\left(x\right)@@ is undefined if @@f(a)\neq L@@

Limit Definition of the Derivative #1

• @@f'ig(xig)=rac{fig(x+hig)-fig(xig)}{h}@@

• @@f'ig(xig)=rac{fig(xig)-fig(x-hig)}{h}@@

• @@f'ig(xig)=rac{fig(x+hig)+fig(xig)}{h}@@

• @@f'ig(xig)=rac{fig(xig)-fig(x+hig)}{h}@@

Average rate of change/slope of the secant line over [a, b]

• @@rac{fig(big)-fig(aig)}{b-a}@@

• @@rac{fig(aig)-fig(big)}{b-a}@@

• @@rac{fig(big)-fig(aig)}{a-b}@@

• @@rac{fig(aig)+fig(big)}{b-a}@@

Limit of a Right Riemann Sum

• @@rac{b-a}{n}@@

• @@ ext{lim}_{n o ext{infinity}} ext{sum}_{k=1}^{n} f(a+krac{b-a}{n})rac{b-a}{n}@@

• @@ ext{lim}_{x o 0} ext{sum}_{k=1}^{n} f(a+krac{b-a}{n})rac{b-a}{n}@@

• @@ ext{lim}_{n o ext{infinity}} ext{sum}_{k=1}^{n} f(a+krac{b-a}{n})rac{b-a}{n}@@

Limit Definition of the Derivative #2

• @@f'ig(aig)= ext{lim}_{x ightarrow a}rac{fig(xig)-fig(aig)}{x-a}@@

• @@f'ig(aig)=rac{fig(xig)-fig(aig)}{x-a}@@

• @@f'ig(aig)= ext{lim}_{x ightarrow a}rac{fig(aig)-fig(xig)}{x-a}@@

• @@f'ig(aig)= ext{lim}_{x ightarrow a}rac{fig(aig)-fig(aig)}{x-a}@@

Definition of Continuity at x = a

• @@f\left(a\right)@@ exists, @@\lim_{x\rightarrow a}f\left(x\right)@@ exists, and @@f\left(a\right)=\lim_{x\rightarrow a}f\left(x\right)@@

• @@f\left(a\right)@@ does not exist, @@\lim_{x\rightarrow a}f\left(x\right)@@ exists, and @@f\left(a\right)\neq\lim_{x\rightarrow a}f\left(x\right)@@

• @@f\left(a\right)@@ exists, @@\lim_{x\rightarrow a}f\left(x\right)@@ does not exist, and @@f\left(a\right)=\lim_{x\rightarrow a}f\left(x\right)@@

• @@f\left(a\right)@@ exists, @@\lim_{x\rightarrow a}f\left(x\right)@@ exists, and @@f\left(a\right)\neq\lim_{x\rightarrow a}f\left(x\right)@@